The volume product of convex bodies with many hyperplane symmetries

Franck Barthe (1) , Matthieu Fradelizi (2)

1 LSP - Laboratoire de Statistique et Probabilités
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

Mahler's conjecture predicts a sharp lower bound on the volume of the polar of a convex body in terms of its volume. We confirm the conjecture for convex bodies with many hyperplane symmetries in the following sense: their hyperplanes of symmetries have a one-point intersection. Moreover, we obtain improved sharp lower bounds for classes of convex bodies which are invariant by certain reflection groups, namely direct products of the isometry groups of regular polytopes.

Domaines: Analyse fonctionnelle [math.FA]

Franck Barthe, Matthieu Fradelizi. The volume product of convex bodies with many hyperplane symmetries. American Journal of Mathematics, 2013, 135 (2), pp.1-37. ⟨10.1353/ajm.2013.0018⟩. ⟨hal-00793781⟩